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Commuting rational functions revisited

Part of: Complex dynamical systems Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  15 August 2019

FEDOR PAKOVICH Open the ORCID record for FEDOR PAKOVICH [Opens in a new window]
FEDOR PAKOVICH*
Affiliation:
Department of Mathematics, Ben Gurion University, P.O. Box 653, Beer Sheva, 8410501, Israel email pakovich@math.bgu.ac.il
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Abstract

Let $B$ be a rational function of degree at least two that is neither a Lattès map nor conjugate to $z^{\pm n}$ or $\pm T_{n}$. We provide a method for describing the set $C_{B}$ consisting of all rational functions commuting with $B$. Specifically, we define an equivalence relation $\underset{B}{{\sim}}$ on $C_{B}$ such that the quotient $C_{B}/\underset{B}{{\sim}}$ possesses the structure of a finite group $G_{B}$, and describe generators of $G_{B}$ in terms of the fundamental group of a special graph associated with $B$.

Keywords

commuting rational functionsthe Ritt theoremcommon iterates

MSC classification

Primary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 1 , January 2021 , pp. 295 - 320
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Copyright
© Cambridge University Press, 2019

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